EPA. Even a short stretch of double-fense in San Diego across the infamously named "Smuggler's Gulch" was blocked by years due to environmental regulation and lawsuits. Perry knows this. Perry also knows 1200 miles of the border is a giant river. If you can't patrol a river, you're doing it wrong. Perry's for a fence where useful but using dollars wisely because manpower and intelligence will still be needed and is far more cost effective.
Giant river, ROFLOL, wrong, many places you can walk across it and only get your ankles wet, but if you are south of El Paso your ankles may get a brown coating on them. Have you never seen it.??
FYI, the Rio Grande is not "giant". For most of the 1200 miles, it's maybe 50' wide and can be easily waded.
From Fort Hancock to Presidio, it's mostly salt cedar and sand...with an occasional puddle.
Your point stands however: The border is a river. And all a fence will do is give Mexico control of the river.
The Israelis know where to build fences. And where not to. There is no fence along the Jordan River, at the border with Jordan. Access to its water is too important -- for both livestock and humans.
As a consequence, the Jordan River border is secured by a patrol road, observation posts (well-armed) and sensors. No fence.